{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T15:01:07Z","timestamp":1774537267081,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,9]],"date-time":"2022-09-09T00:00:00Z","timestamp":1662681600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we discuss the initial state of the universe at the Big Bang. By using the ideas of Freedman in the proof of the disk embedding theorem for 4-manifolds, we describe the corresponding spacetime as a gravitational instanton. The spatial space is a fractal space (wild embedded 3-sphere). Then, we construct the quantum state from this fractal space. This quantum state is part of the string algebra of Ocneanu. There is a link between the Jones polynomial and Witten\u2019s topological field theory. Using this link, we are able to determine the physical theory (action) as the Chern\u2013Simons functional. The gauge fixing of this action determines the foliation of the spacetime and the smoothness properties. Finally, we determine the quantum symmetry of the quantum state to be the enveloped Lie algebra Uq(sl2(C)), where q is the fourth root of unity.<\/jats:p>","DOI":"10.3390\/sym14091887","type":"journal-article","created":{"date-parts":[[2022,9,9]],"date-time":"2022-09-09T04:54:41Z","timestamp":1662699281000},"page":"1887","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Big Bang and Topology"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4813-1010","authenticated-orcid":false,"given":"Torsten","family":"Asselmeyer-Maluga","sequence":"first","affiliation":[{"name":"German Aerospace Center (DLR), Rosa-Luxemburg-Str. 2, 10178 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7296-7355","authenticated-orcid":false,"given":"Jerzy","family":"Kr\u00f3l","sequence":"additional","affiliation":[{"name":"Cognitive Science and Mathematical Modelling Chair, University of Information Technology and Management, ul. Sucharskiego 2, 35-225 Rzesz\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alissa","family":"Wilms","sequence":"additional","affiliation":[{"name":"Physics Department, Freie Universit\u00e4t Berlin, Arnimallee 14, 14195 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2960","DOI":"10.1103\/PhysRevD.28.2960","article-title":"Wave function of the universe","volume":"28","author":"Hartle","year":"1983","journal-title":"Phys. Rev. D"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/S0370-2693(98)00234-2","article-title":"Open inflation without false vacua","volume":"425","author":"Hawking","year":"1998","journal-title":"Phys. Lett. B"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1350055","DOI":"10.1142\/S0219887813500552","article-title":"Quantum geometry and wild embeddings as quantum states","volume":"10","year":"2013","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Freedman, M., and Quinn, F. (1990). Topology of 4-Manifolds, Princeton University Press.","DOI":"10.1515\/9781400861064"},{"key":"ref_5","first-page":"357","article-title":"The topology of four-dimensional manifolds","volume":"17","author":"Freedman","year":"1982","journal-title":"J. Diff. Geom."},{"key":"ref_6","first-page":"647","article-title":"The disk problem for four-dimensional manifolds","volume":"17","author":"Freedman","year":"1983","journal-title":"Proc. Internat. Cong. Math. Warzawa"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Evans, D.E., and Takesaki, M. (1988). Quantized groups, string algebras and Galois theory for algebras. Operator Algebras and Applications, Cambridge University Press.","DOI":"10.1017\/CBO9780511662270"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01389127","article-title":"Index of subfactors","volume":"72","author":"Jones","year":"1983","journal-title":"Invent. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1090\/S0273-0979-1985-15304-2","article-title":"A polynomial invariant for knots via von Neumann algebras","volume":"12","author":"Jones","year":"1985","journal-title":"BAMS"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Witten, E. (1988). 2 + 1 dimensional gravity as an exactly soluble system. Nucl. Phys. B, 311.","DOI":"10.1016\/0550-3213(88)90143-5"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/BF01217730","article-title":"Quantum field theory and the Jones polynomial","volume":"121","author":"Witten","year":"1989","journal-title":"Commun. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1007\/s10714-015-1872-x","article-title":"How to include fermions into general relativity by exotic smoothness","volume":"47","author":"Brans","year":"2015","journal-title":"Gen. Relativ. Grav."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2825","DOI":"10.1007\/s10714-012-1419-3","article-title":"On the geometrization of matter by exotic smoothness","volume":"44","year":"2012","journal-title":"Gen. Rel. Grav."},{"key":"ref_14","unstructured":"Hawking, S.W., and Ellis, G.F.R. (1994). The Large Scale Structure of Space-Time, Cambridge University Press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"5494","DOI":"10.1063\/1.530761","article-title":"Exotic smoothness and physics","volume":"35","author":"Brans","year":"1994","journal-title":"J. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1785","DOI":"10.1088\/0264-9381\/11\/7\/015","article-title":"Localized exotic smoothness","volume":"11","author":"Brans","year":"1994","journal-title":"Class. Quant. Grav."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Asselmeyer-Maluga, T. (2016). Smooth quantum gravity: Exotic smoothness and Quantum gravity. At the Frontiers of Spacetime: Scalar-Tensor Theory, Bell\u2019s Inequality, Mach\u2019s Principle, Exotic Smoothness, Springer.","DOI":"10.1007\/978-3-319-31299-6_15"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1002\/cpa.3160310304","article-title":"On the Ricci curvature of a compact K\u00e4hler manifold and the complex Monge-Amp\u00e8re equation","volume":"31","author":"Yau","year":"1978","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4310\/MRL.1996.v3.n2.a1","article-title":"Four-manifolds without Einstein metrics","volume":"3","author":"LeBrun","year":"1996","journal-title":"Math. Res. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Asselmeyer-Maluga, T., and Brans, C.H. (2008). Exotic Smoothness and Physics, World Scientific.","DOI":"10.1142\/4323"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1250135","DOI":"10.1142\/S0217732312501350","article-title":"On topological restrictions of the spacetime in cosmology","volume":"27","year":"2012","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_22","first-page":"269","article-title":"An application of gauge theory to the topology of 4-manifolds","volume":"18","author":"Donaldson","year":"1983","journal-title":"J. Diff. Geom."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.dark.2017.12.002","article-title":"How to obtain a cosmological constant from small exotic R4","volume":"19","author":"Krol","year":"2018","journal-title":"Phys. Dark Universe"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1950097","DOI":"10.1142\/S0217732319500974","article-title":"A topological approach to neutrino masses by using exotic smoothness","volume":"34","author":"Krol","year":"2019","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_25","unstructured":"Asselmeyer-Maluga, T., and Krol, J. (2018). A topological model for inflation. arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1297","DOI":"10.1090\/S0002-9939-1995-1246517-X","article-title":"An explicit family of exotic Casson handles","volume":"123","author":"Bizaca","year":"1995","journal-title":"Proc. AMS"},{"key":"ref_27","first-page":"458","article-title":"Elliptic surfaces and some simple exotic R4\u2019s","volume":"43","author":"Gompf","year":"1996","journal-title":"J. Diff. Geom."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1350158","DOI":"10.1142\/S0217732313501587","article-title":"Decoherence in quantum cosmology and the cosmological constant","volume":"28","year":"2013","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Hein, H.-J., Sun, S., Viaclovsky, J., and Zhang, R. (2022). Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces. arXiv.","DOI":"10.1090\/jams\/978"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Gompf, R.E., and Stipsicz, A.I. (1999). 4-Manifolds and Kirby Calculus, American Mathematical Society.","DOI":"10.1090\/gsm\/020"},{"key":"ref_31","unstructured":"Hein, H.-J., Sun, S., Viaclovsky, J., and Zhang, R. (2021). Gravitational instantons and del Pezzo surfaces. arXiv."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"012009","DOI":"10.1088\/1742-6596\/1194\/1\/012009","article-title":"Hyperbolic groups, 4-manifolds and quantum gravity","volume":"1194","year":"2019","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_33","unstructured":"Ade, P.A.R., Aghanim, N., Armitage-Caplan, C., Arnaud, M., Ashdown, M., Atrio-Barandela, F., Aumont, J., Baccigalupi, C., Banday, A.J., and Barreiro, R.B. (2013). Planck 2013 results. XVI. cosmological parameters. arXiv."},{"key":"ref_34","unstructured":"Ade, P.A.R., Aghanim, N., Armitage-Caplan, C., Arnaud, M., Ashdown, M., Atrio-Barandela, F., Aumont, J., Baccigalupi, C., Banday, A.J., and Barreiro, R.B. (2013). Planck 2013 results. XXII. constraints on inflation. arXiv."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2899","DOI":"10.1088\/0264-9381\/11\/12\/007","article-title":"Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories","volume":"11","author":"Connes","year":"1994","journal-title":"Class. Quan. Grav."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3604","DOI":"10.1063\/1.533323","article-title":"On revolutionizing quantum field theory with Tomita\u2019s modular theory","volume":"41","author":"Borchers","year":"2000","journal-title":"J. Math. Phys."},{"key":"ref_37","unstructured":"Asselmeyer-Maluga, T., and Kr\u00f3l, J. (2010). Exotic smooth R4, noncommutative algebras and quantization. arXiv."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Goodman, F., Harpe, P.d., and Jones, V. (1989). Coxeter Graphs and Towers of Algebras, Springer. MSRI Publications Edition.","DOI":"10.1007\/978-1-4613-9641-3"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Asselmeyer-Maluga, T. (2019). Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics. Symmetry, 11.","DOI":"10.3390\/sym11101298"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"335","DOI":"10.2969\/jmsj\/03820335","article-title":"A recursive calculation of the Arf invariant of a link","volume":"38","author":"Murakami","year":"1986","journal-title":"J. Math. Soc. Jpn."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1007\/BF01239527","article-title":"Invariants of three-manifolds via link polynomials and quantum groups","volume":"103","author":"Reshetikhin","year":"1991","journal-title":"Inv. Math."},{"key":"ref_42","unstructured":"Kauffman, L.H. (1999). Functional integration and the Kontsevich integral. Yang-Baxter Systems, Nonlinear Models and Their Applications, Proceedings of the APCTP-Nankai Symposium, Singapore, World Scientific."},{"key":"ref_43","first-page":"A92","article-title":"Un invariant des feuilletages de codimension","volume":"273","author":"Godbillon","year":"1971","journal-title":"C. R. Acad. Sci. Paris Ser. A-B"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1090\/S0273-0979-1984-15301-1","article-title":"Secondary classes and transverse measure theory of a foliation","volume":"11","author":"Hurder","year":"1984","journal-title":"BAMS"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1090\/S0002-9904-1972-12975-6","article-title":"Noncobordant foliations of S3","volume":"78","author":"Thurston","year":"1972","journal-title":"BAMS"},{"key":"ref_46","first-page":"521","article-title":"A survey of foliations and operator algebras","volume":"38","author":"Connes","year":"1984","journal-title":"Proc. Symp. Pure Math."},{"key":"ref_47","unstructured":"Connes, A. (1994). Non-Commutative Geometry, Academic Press."},{"key":"ref_48","first-page":"706","article-title":"A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture","volume":"44","author":"Morgan","year":"1996","journal-title":"J. Diff. Geom."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"915","DOI":"10.4310\/MRL.1997.v4.n6.a11","article-title":"Product formulas along t3 for Seiberg-Witten invariants","volume":"4","author":"Morgan","year":"1997","journal-title":"Math. Res. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Chandrasekaran, V., Longo, R., Penington, G., and Witten, E. (2022). An Algebra of Observables for de Sitter Space. arXiv.","DOI":"10.1007\/JHEP02(2023)082"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1887\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:28:07Z","timestamp":1760142487000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1887"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,9]]},"references-count":50,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["sym14091887"],"URL":"https:\/\/doi.org\/10.3390\/sym14091887","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,9]]}}}